On Whitham's conjecture of a highest cusped wave for a nonlocal dispersive equation

We consider the Whitham equation $u_t + 2u u_x+Lu_x = 0$, where L is the nonlocal Fourier multiplier operator given by the symbol $m(ξ) = \sqrt{\tanh ξ/ξ}$. G. B. Whitham conjectured that for this equation there would be a highest, cusped, travelling-wave solution. We find this wave as a limiting case at the end of the main bifurcation curve of $P$-periodic solutions, and give several qualitative properties of it, including its optimal $C^{1/2}$-regularity. An essential part of the proof consists in an analysis of the integral kernel corresponding to the symbol $m(ξ)$, and a following study of the highest wave. In particular, we show that the integral kernel corresponding to the symbol $m(ξ)$ is completely monotone, and provide an explicit representation formula for it.

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